Growth of regular partitions 3: strong regularity and the vertex partition

We consider here the strong regularity for $3$-uniform hypergraphs developed by Frankl, Gowers, Kohayakawa, Nagle, Rödl, Skokan, and Schacht. This type of regular decomposition comes with two components, a partition of the vertices, and a partition of the pairs of vertices. The data of a regular decomposition also includes two parameters measuring quasirandomness, a fixed constant ${\epsilon }_1>0$, and a function ${\epsilon }_2:\mathbb{N}\to (0,1]$. We define two growth functions associated to a hereditary property $\mathcal{H}$ of $3$-uniform hypergraphs: $T_{\mathcal{H}}({\epsilon }_1,{\epsilon }_2)$ which measures the size of the vertex component, and $L_{\mathcal{H}}(\epsilon ,{\epsilon }_2)$ which measures the size of the pairs component. We introduce the following question. What are the possible asymptotic growth rates of functions of the form $T_{\mathcal{H}}$ and $L_{\mathcal{H}}$? In this paper, we consider this question for $T_{\mathcal{H}}$, proving a separation into four classes: constant, polynomial, exponential, or at least wowzer. The separations among the constant, polynomial and exponential ranges require only slow growing (namely polynomial) choices for ${\epsilon }_2$. The jump to the wowzer range uses a very fast growing ${\epsilon }_2$ and makes crucial use of a lower bound construction for strong graph regularity due to Conlon and Fox.